Appell–Humbert Theorem
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Appell–Humbert theorem describes the
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
s on a
complex torus In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
or complex
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...
. It was proved for 2-dimensional tori by and , and in general by


Statement

Suppose that T is a complex torus given by V/\Lambda where \Lambda is a lattice in a complex vector space V. If H is a
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear map, linear in each of its arguments, but a sesquilinear f ...
on V whose imaginary part E = \text(H) is integral on \Lambda\times\Lambda, and \alpha is a map from \Lambda to the unit circle U(1) = \, called a semi-character, such that :\alpha(u+v) = e^\alpha(u)\alpha(v)\ then : \alpha(u)e^\ is a 1-
cocycle In mathematics a cocycle is a closed cochain (algebraic topology), cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in gr ...
of \Lambda defining a line bundle on T. For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus
\text_(\Lambda,U(1)) \cong \mathbb^/\mathbb^
if \Lambda \cong \mathbb^ since any such character factors through \mathbb composed with the exponential map. That is, a character is a map of the form
\text(2\pi i \langle l^*, -\rangle )
for some covector l^* \in V^*. The periodicity of \text(2\pi i f(x)) for a linear f(x) gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus. Explicitly, a line bundle on T = V/\Lambda may be constructed by
descent Descent may refer to: As a noun Genealogy and inheritance * Common descent, concept in evolutionary biology * Kinship, one of the major concepts of cultural anthropology **Pedigree chart or family tree **Ancestry **Lineal descendant **Heritage ** ...
from a line bundle on V (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms u^*\mathcal_V \to \mathcal_V, one for each u \in \Lambda. Such isomorphisms may be presented as nonvanishing holomorphic functions on V, and for each u the expression above is a corresponding holomorphic function. The Appell–Humbert theorem says that every line bundle on T can be constructed like this for a unique choice of H and \alpha satisfying the conditions above.


Ample line bundles

Lefschetz proved that the line bundle L, associated to the Hermitian form H is ample if and only if H is positive definite, and in this case L^ is very ample. A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on \Lambda\times\Lambda


See also

*
Complex torus In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
for a treatment of the theorem with examples


References

* * * * * {{DEFAULTSORT:Appell-Humbert theorem Abelian varieties Theorems in algebraic geometry Theorems in complex geometry